Weyl-type bounds for Steklov eigenvalues

TL;DR

This paper derives sharp upper and lower bounds for Steklov eigenvalues in smooth domains, linking them to Laplacian eigenvalues on the boundary, and provides asymptotically precise estimates for heat kernel traces.

Contribution

It introduces a novel comparison technique between Steklov and Laplacian eigenvalues using Pohozaev identities, leading to sharp bounds consistent with Weyl asymptotics.

Findings

Established sharp bounds for Steklov eigenvalues and Riesz-means.

Connected Steklov eigenvalues to boundary Laplacian eigenvalues.

Provided asymptotically sharp heat kernel trace estimates.

Abstract

We present upper and lower bounds for Steklov eigenvalues for domains in $\mathbb{R}^{N+1}$ with $C^2$ boundary compatible with the Weyl asymptotics. In particular, we obtain sharp upper bounds on Riesz-means and the trace of corresponding Steklov heat kernel. The key result is a comparison of Steklov eigenvalues and Laplacian eigenvalues on the boundary of the domain by applying Pohozaev-type identities on an appropriate tubular neigborhood of the boundary and the min-max principle. Asymptotically sharp bounds then follow from bounds for Riesz-means of Laplacian eigenvalues.